We prove that if T1, . . . , Tn is a sequence of bounded degree trees so that Ti has i vertices, then Kn has a decomposition into T1, . . . , Tn. This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first o(n) trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our proofs involve Szemerédi's regularity lemma, results on Hamilton decompositions of robust expanders, random walks, iterative absorption as well as a recent blow-up lemma for approximate decompositions.
We prove χ s (G) 1.93Δ(G) 2 for graphs of sufficiently large maximum degree where χ s (G) is the strong chromatic index of G. This improves an old bound of Molloy and Reed. As a by-product, we present a Talagrand-type inequality where we are allowed to exclude unlikely bad outcomes that would otherwise render the inequality unusable.
We prove χ2 for graphs of sufficiently large maximum degree where χ ′ s (G) is the strong chromatic index of G. This improves an old bound of Molloy and Reed. As a by-product, we present a Talagrandtype inequality where it is allowed to exclude unlikely bad outcomes that would otherwise render the inequality unusable.
We prove that for every graph, any vertex subset S, and given integers k,ℓ: there are k disjoint cycles of length at least ℓ that each contain at least one vertex from S, or a vertex set of size O(ℓ·klogk) that meets all such cycles. This generalizes previous results of Fiorini and Herinckx and of Pontecorvi and Wollan.
In addition, we describe an algorithm for our main result that runs in O(klogk·s2·false(f(ℓ)·n+mfalse)) time, where s denotes the cardinality of S.
A (Γ1, Γ2)-labeled graph is an oriented graph with its edges labeled by elements of the direct sum of two groups Γ1, Γ2. A cycle in such a labeled graph is (Γ1, Γ2)-non-zero if it is non-zero in both coordinates. Our main result is a generalization of the Flat Wall Theorem of Robertson and Seymour to (Γ1, Γ2)-labeled graphs. As an application, we determine all canonical obstructions to the Erdős-Pósa property for (Γ1, Γ2)-non-zero cycles in (Γ1, Γ2)-labeled graphs. The obstructions imply that the halfintegral Erdős-Pósa property always holds for (Γ1, Γ2)-non-zero cycles.Moreover, our approach gives a unified framework for proving packing results for constrained cycles in graphs. For example, as immediate corollaries we recover the Erdős-Pósa property for cycles and S-cycles and the half-integral Erdős-Pósa property for odd cycles and odd S-cycles. Furthermore, we recover Reed's Escher-wall Theorem.We also prove many new packing results as immediate corollaries. For example, we show that the half-integral Erdős-Pósa property holds for cycles not homologous to zero, odd cycles not homologous to zero, and Scycles not homologous to zero. Moreover, the (full) Erdős-Pósa property holds for S1-S2-cycles and cycles not homologous to zero on an orientable surface. Finally, we also describe the canonical obstructions to the Erdős-Pósa property for cycles not homologous to zero and for odd S-cycles.
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